Sampled-Data Consensus over Random Networks

TL;DR

This paper investigates the conditions under which consensus can be achieved in networks with random interactions and sampled-data control, establishing equivalences among different types of consensus and deriving critical bounds for inter-sampling intervals.

Contribution

It provides necessary and sufficient conditions for mean-square consensus in random networks and links these to spectral properties and sampling intervals, extending understanding of stochastic network consensus.

Findings

Consensus in expectation, mean square, and almost surely are equivalent under certain conditions.

Critical inter-sampling interval bounds determine whether the system converges or diverges.

Numerical simulations validate the theoretical conditions for consensus and divergence.

Abstract

This paper considers the consensus problem for a network of nodes with random interactions and sampled-data control actions. We first show that consensus in expectation, in mean square, and almost surely are equivalent for a general random network model when the inter-sampling interval and network size satisfy a simple relation. The three types of consensus are shown to be simultaneously achieved over an independent or a Markovian random network defined on an underlying graph with a directed spanning tree. For both independent and Markovian random network models, necessary and sufficient conditions for mean-square consensus are derived in terms of the spectral radius of the corresponding state transition matrix. These conditions are then interpreted as the existence of critical value on the inter-sampling interval, below which global mean-square consensus is achieved and above which the system diverges in mean-square sense for some initial states. Finally, we establish an upper bound on the inter-sampling interval below which almost sure consensus is reached, and a lower bound on the inter-sampling interval above which almost sure divergence is reached. Some numerical simulations are given to validate the theoretical results and some discussions on the critical value of the inter-sampling intervals for the mean-square consensus are provided.